Techniques for mitigating the checkerboard formation: application in bone remodeling simulations

Abstract

The checkerboard phenomenon is a numerical problem that can occur in bone remodeling simulations. This phenomenon degrades the quality of the results and emphasizes a strong mesh dependency. The node-based approach is a well-used method for solving this problem. However, this method does not provide a stable solution, and long-time simulations do not reach steady-state. This paper aims to discuss techniques for mitigating the checkerboard formation and obtaining bone characterization. For this, we apply the simple nodal average technique on stress, density, or both simultaneously. Then, we compare the element-based and node-based approaches with and without smoothing. Also, we introduced a quantification parameter called standard density variation. This parameter measures the intensity of the checkerboard formation and analyzes the evolution of the bone tissue behavior. The application of the smoothing techniques mitigates the checkerboard formation. Also, the application allows obtaining a stable behavior for long-time simulations. The standard density variation allowed visualizing this stable behavior. In particular, the stress field smoothing technique provides the best results. Besides the stabilization and checkerboard control, the technique allows obtaining a physiological density distribution.

Keywords: Checkerboard mitigation; Convergence; Finite element method; Phenomenological model; Smoothing techniques.